Assosiated Legendre function of the second kind of type 3: Representations through more general functions (formula 07.12.26.0064)

LegendreQ

$Mathematica Notation$

Hypergeometric Functions

LegendreQ[nu,mu,3,z]

Representations through more general functions

Through Meijer G

Generalized cases involving products of Legendre Q

http://functions.wolfram.com/07.12.26.0064.01

Input Form

(1/Sqrt[z^2 + 1]) LegendreQ[\[Nu], \[Mu], 3, Sqrt[z^2 + 1]/z] LegendreQ[\[Nu], \[Mu] + 1, 3, Sqrt[z^2 + 1]/z] == (-((E^(2 Pi I \[Mu]) Sqrt[Pi] Gamma[1 + \[Mu] + \[Nu]])/ (2 Gamma[1 - \[Mu] + \[Nu]]))) MeijerG[{{1, 1 + \[Mu], -\[Mu]}, {}}, {{\[Nu] + 1}, {1/2, -\[Nu]}}, z, 1/2] /; !IntervalMemberQ[Interval[{-Infinity, 0}], z]

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> 𝔔 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mfrac> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mi> z </mi> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]], "z"], HoldComplete[LegendreQ, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> 𝔔 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <semantics> <mfrac> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mi> z </mi> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]], "z"], HoldComplete[LegendreQ, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> μ </mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "3"]], RowBox[List["1", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["1", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Mu]", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Mu]"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; 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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]], "z_"]]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["\[Mu]_", "+", "1"]], ",", "3", ",", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]], "z_"]]], "]"]]]], SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Mu]"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "\[Mu]"]], ",", RowBox[List["-", "\[Mu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", "\[Nu]"]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29

LegendreQ[nu,z]
LegendreQ[nu,mu,z]
LegendreQ[nu,mu,2,z]